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(-1.0, 5.0)
Ingeniería Biomédica
2024-08-12
(-1.0, 6.0)
(-1.0, 5.0)
Zero Mean (Admissibility Condition)
The function must have an average value of zero. Mathematically, this is expressed as:
\[\int_{-\infty}^{\infty} \psi(t) \, dt = 0\]
This condition ensures that the wavelet can detect changes or “details” in the signal rather than its average or constant components.
Square Integrability
The function \(\psi(t)\) must be square integrable, meaning it has finite energy:
\[\int_{-\infty}^{\infty} |\psi(t)|^2 \, dt < \infty\]
This requirement ensures that the wavelet’s energy is finite, making it possible to localize the function in both time and frequency domains. Functions that satisfy this belong to the \(L^2(\rm I\!R)\) space, which is the space of all functions with finite energy.
Admissibility Constant
The wavelet’s Fourier transform, \(\hat{\psi}(\omega)\), should satisfy the admissibility condition:
\[C_\psi = \int_{-\infty}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{|\omega|} \, d\omega < \infty\]
where \(\hat{\psi}(\omega)\) is the Fourier transform of \(\psi(t)\), and \(\omega\) represents angular frequency. This condition implies that \(\hat{\psi}(\omega)\) must approach zero as \(\omega \rightarrow 0\) meaning the wavelet has no component at zero frequency (or DC component). This condition is crucial for ensuring that the wavelet transform is invertible.
Oscillatory Nature
A mother wavelet should generally be oscillatory or “wavelike” (hence the term “wavelet”). This oscillatory behavior allows the wavelet to capture variations in the signal. For example, wavelets like the Morlet wavelet resemble decaying sinusoids. This oscillatory nature helps the wavelet capture both high-frequency and low-frequency components effectively.
Compact Support
Although not strictly necessary, compact support is often a desirable property. Compact support means that the function is non-zero only over a finite interval, making it well-localized in time. This allows for efficient computation and good localization in the time domain. For example, the Haar wavelet has compact support, while others, like the Morlet wavelet, do not have strict compact support but still decay rapidly.
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The continuous wavelet transform of a signal \(f(t)\) is defined as:
\[W_f(a, b) = \int_{-\infty}^{\infty} f(t) \, \psi^*\left(\frac{t - b}{a}\right) \, dt\]
where:
The continuous wavelet transform of a signal \(f(t)\) is defined as:
\[W_f(a, b) = \int_{-\infty}^{\infty} f(t) \, \psi^*\left(\frac{t - b}{a}\right) \, dt\]
where:
The inverse continuous wavelet transform is given by:
\[f(t) = \frac{1}{C_{\psi}} \int_{0}^{\infty} \int_{-\infty}^{\infty} W_f(a, b) \, \psi\left(\frac{t - b}{a}\right) \frac{db \, da}{a^2}\]
where:
where \(C_{\psi}\) is a normalization constant, defined as:
\[C_{\psi} = \int_{0}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{\omega} \, d\omega\]
and \(\hat{\psi}(\omega)\) is the Fourier transform of the wavelet \(\psi(t)\).
The discrete wavelet transform decomposes the signal at discrete levels of scale. For a signal \(x[n]\), the wavelet decomposition is defined as:
\[c_{j, k} = \sum_{n} x[n] \, \psi_{j, k}(n)\]
where:
The decomposition typically consists of approximation and detail coefficients at each scale:
Approximation coefficients \(a_j\): capture the low-frequency components. Detail coefficients \(d_j\) capture the high-frequency components.
The inverse discrete wavelet transform reconstructs the original signal from its approximation and detail coefficients: \[x[n] = \sum_{j} \sum_{k} c_{j, k} \, \psi_{j, k}(n)\]
This reconstruction process involves upsampling and filtering of the approximation and detail coefficients at each scale.
Use CWT when:
Use DWT when: